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LectureFeb29_MATH4321_12S

# LectureFeb29_MATH4321_12S - Recall that Equilibrium...

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Recall that Equilibrium Principle: BR to each other Maximin Principle: Safety first For Player I: Find p so that Min q p T Aq is largest. p is called the Safety Strategy or Optimal Strategy. Min q p T Aq is the lower value. For Player II: Find q so that Max p p T Aq is smallest. q is called the Safety Strategy or Optimal Strategy. Max p p T Aq is the upper value. Minimax Theorem: Maximin=Minimax

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Theorem : (Maximin Principle Equilibrium Principle) Let p ~ belong to X *, q ~ belong to Y *, be safety strategies for Player I and II respectively. Then, p ~ , q ~ are best responses to each other i.e. <p ~ , q ~> is an equilibrium pair.
Proof: p ~ is a safety strategy for Player I means Min q p ~T Aq = Max p Min q p T Aq q ~ is a safety strategy for Player II means Max p p T Aq ~ = Min q Max p p T Aq

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Then, we show in the following p ~ is a BR to q ~ . p ~T Aq ~ Min q p ~T Aq = Max p Min q p T Aq = Min q Max p p T Aq = Max p p T Aq ~ p T Aq ~ , for any p in X * This completes the proof. Similarly, q ~ is a BR to p ~ .
Theorem (Equilibrium Principle Maximin Principle): Let p ~ , q ~ be best responses to each other i.e. <p ~ , q ~ > is an equilibrium pair. Then, p ~ (in X * ), q ~ (in Y * )are safety strategies for Player I and II respectively.

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Proof: As p ~ is a BR to q ~ , we have p ~T Aq ~ = Max p p T Aq ~ Max p Min q p T Aq = Min q Max p p T Aq Min q p ~T Aq = p ~T Aq ~ (q ~ is a BR to p ~ ) Thus, Min q p ~T Aq = Max p Min q p T Aq and p ~ is a safety strategy for Player I. Similarly, q ~ is a safety strategy for Player II.

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Theorem: Let <p 1 , q 1 >, <p 2 , q 2 > be equilibrium pairs. Then, <p 1 , q 2 > is also an equilibrium pair.
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